Rotation, reflection, and symmetry rules
What this domain tests
Items built on symmetry ask you to compare, complete, or count symmetric properties: axes of symmetry, order of rotational symmetry, and the effect of reflecting a figure in a stated line. The exam expects exact counts, not approximate visual judgements.
Core facts
- Order of rotational symmetry is 360 divided by the smallest angle that maps the figure onto itself: square 4, equilateral triangle 3, regular pentagon 5, regular hexagon 6, rectangle 2, circle unlimited.
- Axes of symmetry: square 4, equilateral triangle 3, regular pentagon 5, regular hexagon 6, rectangle 2 (vertical and horizontal, not the diagonals), isosceles triangle 1, parallelogram 0.
- Reflection reverses orientation; rotation does not.
- Two successive reflections in lines meeting at angle x equal a rotation of 2x about the intersection point. Vertical then horizontal mirrors give a 180-degree rotation.
- A figure can be rotationally symmetric without any mirror symmetry, as a swastika-like or comma-shaped motif shows.
Worked example
A motif is reflected first in a vertical axis, then in a horizontal axis. The two mirror lines meet at 90 degrees, so the combined effect is a rotation of 180 degrees, not 90. The third frame must therefore look like the first turned upside down, with any asymmetric detail, such as a dot near the top-left, moving to the bottom-right. Options showing a 90-degree turn or a single reflection are wrong even though they look plausible for a nearly symmetric motif.
Traps that cost marks
- Counting the diagonals of a rectangle as axes of symmetry; they are not.
- Confusing order of rotational symmetry with number of axes; they coincide for regular polygons but not in general.
- Judging symmetry from the outline while internal detail breaks it, or the reverse.
- Mixing up the mirror line: a horizontal axis flips top to bottom, not left to right.
- Assuming that reflection and rotation commute. Applying a reflection and then a rotation gives a different figure from the reverse order.